On a generalization of both affine planes and MDS-codes with applications to experiment planning
type: Item
instance of: scholarly article
title: On a generalization of both affine planes and MDS-codes with applications to experiment planning
zbMATH Open document ID: 0889.51017
DOI: 10.1007/BF01229571
author: Jörn Quistorff
published in: Journal of Geometry
publication date: 1998-06-08T00:00:00Z
review text: In 1986, \textit{W. Benz} [Ann. Discrete Math. 30, 15-30 (1986; Zbl 0591.51017)] introduced the very general concept of a \(T(t,q,r,n)\) structure, which, by various choices of the parameters, may be a 3-net, an affine plane, a Laguerre plane, a Minkowski plane, an MDS-code or certain generalizations of these structures. In 1992, \textit{W. Benz} [Ann. Discrete Math. 52, 25-36 (1992; Zbl 0768.51012)] reformulated the definition, basing it on the concept of a \((q,r,n)\)-matrix. Given a set \(P\), with \(|P|= qr\), a \((q,r,n)\)-matrix is an \(n \times r\) matrix whose entries are \(q\)-subsets of \(P\), such that each row is a partition of \(P\) and the cardinality of the intersection of any \(j\) \( (1 \leq j \leq n )\) entries, with no two entries selected from the same row, is independent of the choice of entries. In the original formulation, \(n\) \(q \times r\) matrices, whose entries are elements of \(P\), were used. Each of these, upon amalgamating the columns into sets, becomes a row of a \((q,r,n)\)-matrix in the new formulation. The author of the work under review provides a simplification of the definition of a \((q,r,n)\)-matrix by showing that in the definition it suffices to consider only sets of \(n\) entries, no two from the same row. He then proceeds to taking the amalgamation process one step further, removing the matrix structure from a \((q,r,n)\)-matrix (and hence, the implicit orderings imposed by that structure) by defining a \((q,r,n)-set\) as a set of \(n\) partitions of \(P\) with the appropriate properties. This leads to no additional clarity or applicability of the underlying concept. An \(r\)-set \(X\) of \(P\) is called an interaction of a \((q,r,n)\)-matrix (or set) if every component of every partition meets \(X\) (in a unique point). Let \(M\) be a \((q,r,n)\)-matrix and \(B\) a set of interactions of \(M\). For \(t \leq r\), \((M,B)\) is called a \(T(t,q,r,n)\)-structure if, and only if, for every \(t\)-subset \(\Delta\) of \(P\) wi…
Mathematics Subject Classification ID: 51E25
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 1053895
zbMATH Keywords: \(T\)-structures
zbMATH Keywords: \(T(t,q,r,n)\) structures
zbMATH Keywords: \((q,r,n)\)-matrices
zbMATH Keywords: affine planes
zbMATH Keywords: Laguerre planes
zbMATH Keywords: Minkowski planes
zbMATH Keywords: near-fields
reviewed by: William E. Cherowitzo
MaRDI profile type: Publication
cites work: Q3719380
cites work: Q4028416
cites work: Q4135542
cites work: Q5660789
cites work: Q4200788
cites work: Q4836745
full work available at URL: https://doi.org/10.1007/bf01229571
OpenAlex ID: W2146628397
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description: scientific article; zbMATH DE number 1053895